An algebra with the identityWe describe bases of free right-symmetric algebras and free Novikov algebras and give realizations of them in terms of trees. The free Lie algebra is obtained as a Lie subalgebra of the free right-symmetric algebra. We use our methods to study identities of Witt algebras.
Nonassociative algebras satisfying the polynomial identities x 1 (x 2 x 3) = x 2 (x 1 x 3) and (x 1 x 2)x 3 = (x 1 x 3)x 2 are called bicommutative. We prove the following results: (i) Finitely generated bicommutative algebras are weakly noetherian, i.e., satisfy the ascending chain condition for two-sided ideals. (ii) We give the positive solution to the Specht problem (or the finite basis problem) for varieties of bicommutative algebras over an arbitrary field of any characteristic.
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